Download Fitting Selected Random Planetary Systems to Titius

ICARUS
135, 549–557 (1998)
IS985999
ARTICLE NO.
Fitting Selected Random Planetary Systems to Titius–Bode Laws
Wayne Hayes
Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, Ontario M5S 3G4, Canada
E-mail: [email protected]
and
Scott Tremaine
Canadian Institute for Theoretical Astrophysics and Canadian Institute for Advanced Research Program in Cosmology and
Gravity, McLennan Labs, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada
Received October 10, 1997; revised June 30, 1998
Key Words: planetary dynamics; orbits; planetary formation;
celestial mechanics; computer techniques.
Simple ‘‘solar systems’’ are generated with planetary orbital
radii r distributed uniformly random in log r between 0.2 and 50
AU, with masses and order identical to our own Solar System. A
conservative stability criterion is imposed by requiring that
adjacent planets are separated by a minimum distance of k
times the sum of their Hill radii for values of k ranging from
0 to 8. Least-squares fits of these systems to generalized Bode
laws are performed and compared to the fit of our own Solar
System. We find that this stability criterion and other ‘‘radiusexclusion’’ laws generally produce approximately geometrically
spaced planets that fit a Titius–Bode law about as well as our
own Solar System. We then allow the random systems the same
exceptions that have historically been applied to our own Solar
System. Namely, one gap may be inserted, similar to the gap
between Mars and Jupiter, and up to 3 planets may be ‘‘ignored,’’ similar to how some forms of Bode’s law ignore Mercury, Neptune, and Pluto. With these particular exceptions, we
find that our Solar System fits significantly better than the
random ones. However, we believe that this choice of exceptions, designed specifically to give our own Solar System a
better fit, gives it an unfair advantage that would be lost if
other exception rules were used. We compare our results to
previous work that uses a ‘‘law of increasing differences’’ as a
basis for judging the significance of Bode’s law. We note that
the law of increasing differences is not physically based and is
probably too stringent a constraint for judging the significance
of Bode’s law. We conclude that the significance of Bode’s law
is simply that stable planetary systems tend to be regularly
spaced and conjecture that this conclusion could be strengthened by the use of more rigorous methods of rejecting unstable
planetary systems, such as long-term orbit integrations.  1998
Academic Press
‘‘For a statistician, fitting a three-parameter curve of uncertain form to ten points with three exceptions certainly brings
one to the far edge of the known world.’’
— Bradley Efron (1971)
1. INTRODUCTION
The Titius–Bode ‘‘law,’’
ri 5 0.4 1 0.15 3 2i,
i 5 2y, 1, ... , 8,
(1)
roughly describes the planetary semi-major axes in astronomical units (AU), with Mercury assigned i 5 2y, Venus
i 5 1, Earth i 5 2, etc. Usually the asteroid belt is counted
as i 5 4. The law fits the planets Venus through Uranus
quite well, and successfully predicted the existence and
locations of Uranus and the asteroids. However, (i) the
law breaks down badly for Neptune and Pluto; (ii) there
is no reason why Mercury should have i 5 2y rather than
i 5 0, except that it fits better that way; (iii) the total mass
of the asteroid belt is far smaller than the mass of any
planet, so it is not clear that it should be counted as one.
The question of the significance of Bode’s law has taken
on increased interest with discoveries of extra-solar planets
and is also worth reexamination because computer speeds
now permit more powerful statistical tests than were previously possible.
A partial history of the law and attempts to explain it
up to the year 1971 can be found in Nieto (1972). Most
modern arguments concerning the validity of Bode’s law
can be assigned to one of three broad classes:
1. Attempts to elucidate the physical processes leading
to Bode’s law. These are based on a variety of mechanisms,
including dynamical instabilities in the protoplanetary disk
(Graner and Dubrulle 1994, Dubrulle and Graner 1994,
Li et al. 1995), gravitational interactions between planetesi-
549
0019-1035/98 $25.00
Copyright  1998 by Academic Press
All rights of reproduction in any form reserved.
550
HAYES AND TREMAINE
mals (Lecar 1973), or long-term instabilities of the planetary orbits (Hills 1970, Llibre and Piñol 1987, Conway and
Elsner 1988). We shall not comment on these explanations,
except to say that we find none of them entirely convincing.
2. Discussions that ignore physics but try to assess
whether the success of Bode’s law is statistically significant:
Good (1969) performs a likelihood test under the null
hypothesis that the planet distances should be distributed
uniformly random in log r. He includes the asteroid belt,
but ignores Mercury, Neptune, and Pluto, subjectively assigning (i.e., guessing) a factor of 5 penalty to his likelihood
ratio for ignoring these planets. He concludes that there
is a likelihood ratio of 300–700 in favor of Bode’s law
being ‘‘real’’ rather than artifactual. Efron (1971) attacks
Good’s analysis, in particular his choice of null hypothesis.
(Good’s and Efron’s articles are followed by over a dozen
extended ‘‘comments’’ from other statisticians.) Efron
notes that the difference between semi-major axes of adjacent planets is an increasing function of distance for all
adjacent planet pairs except Neptune–Pluto. He proposes,
without physical basis, that this law of increasing differences
is a better null hypothesis, the only reason cited being that
Bode’s law ‘‘predicts’’ increasing differences. Duplicating
Good’s analysis with this new null hypothesis, he computes
a likelihood ratio in favor of Bode’s law of only 8:5 and
concludes that ‘‘there is no compelling evidence for believing that Bode’s law is not artifactual.’’ Conway and
Zelenka (1988) repeat Efron’s analysis using the law of
increasing differences, this time ignoring only Pluto, and
computing a more realistic penalty for doing so. They compute a likelihood ratio of approximately unity, also concluding that Bode’s law is artifactual. We believe that these
analyses are flawed because there is no physical basis for
the law of increasing differences; in fact later we will show
that systems that are stable according to our criteria only
rarely satisfy the law of increasing differences.
3. Discussions of other laws that may influence the spacing of the planets. Many of these involve resonances between the mean motions of the planets, such as Molchanov
(1968; but see Hénon 1969), Birn (1973), and Patterson
(1987). A promising development is the recognition that
planets are capable of migrating significant distances after
their formation (Fernández and Ip 1984, Wetherill 1988,
Ipatov 1993, Lin et al. 1996, Trilling et al. 1998). For example, this process can explain the resonant relationship between Neptune and Pluto (Malhotra 1993, 1995) and may
explain the spacing of the terrestrial planets (Laskar 1997).
The present paper combines the first two of these approaches: we generate a broad range of possible model
planetary systems and exclude those that are known to be
dynamically unstable. We then ask which of the remaining
ones satisfy laws similar to Bode’s.
2. METHOD
2.1. Radius-Exclusion Laws
A necessary, but not sufficient, condition for the stability
of a planetary system is that its planets never get ‘‘too
close to each other’’ (Lecar 1973). This can be formalized
into several ‘‘radius-exclusion laws.’’
1. Simple scaling arguments for near-circular, coplanar
orbits suggest that a test particle on a stable orbit cannot
approach a planet more closely than k Hill radii for some k,
using the Hill radius h as defined by Lissauer (1987, 1993),
S D
1
h 5 HMr,
HM 5
M 3
,
3M(
(2)
for a planet of mass M, semi-major axis r, and fractional
Hill radius HM . We shall extend this criterion to two adjacent planets with nonzero mass by summing their respective Hill radii. There are other plausible ways of combining
adjacent planets: it might be more physically reasonable
to use the sum of the masses to define a single combined
Hill radius, although it is not clear that this is preferred
when more than two planets are involved. Exponents other
than 1/3 may also be reasonable (Wisdom 1980, Chambers
et al. 1996). However, the difference between these approaches is probably unimportant given the uncertainty in
k, as discussed below.
2. For noncircular orbits, we also expect that the aphelion distance of the inner planet is less than the perihelion
distance of the outer one. In other words, if the ith planet
has semi-major axis ri and eccentricity ei , we expect that
ri (1 1 ei) , ri11(1 2 ei11). Taking this further, we may
demand (very conservatively) that the planets are separated by a Hill radius even at their closest possible approach, giving
ri (1 1 ei 1 HMi ) , ri11(1 2 ei11 2 HMi11).
3. Several authors have argued that boundaries between
stable and unstable orbits occur at resonances of the form
j:( j 1 1) (Birn 1973, Wisdom 1980, Weidenschilling and
Davis 1985, Patterson 1987, Holman and Murray 1996).
Weidenschilling and Davis (1985) argue that two planets
are unlikely to form closer than their mutual 2:3 resonance
(because small solid bodies are trapped in the outer j:( j 1
1) resonances of a protoplanet due to gas-induced drag;
once trapped, their eccentricities are pumped up, causing
orbit crossing). For the 2:3 resonance, we can define H2:3
by using Kepler’s third law to define R2:3 5 (3/2)2/3, and
splitting the distance between two adjacent planets by solving R2:3 5 1 1 H2:3 1 R2:3 H2:3 .
551
RANDOM PLANETARY SYSTEMS AND TITIUS–BODE LAWS
Combining all three of Hill radii, eccentricities, and the
2:3 resonance, we obtain
ri (1 1 Vi ) , ri11(1 2 Vi11),
Vi 5 max (H2:3 , ei 1 HMi ).
Some planetary systems in this paper were generated using
planetary masses and ordering of these masses identical
to our own Solar System, while others used equal fractional
radius exclusion for all planets.
Clearly our results will be highly dependent upon the
extent of radius exclusion. For the Hill radius of Eq. (2),
which was derived for the case of two small planets orbiting
a massive central object, a value of k 5 2–4 is believed to
leave the two planets in permanently stable orbits (Wetherill and Cox 1984, 1985, Lissauer 1987, Wetherill 1988, Gladman 1993). For more than two planets, recent work by
Chambers et al. (1996) suggests that no value of k gives
permanent stability. Instead, the stability time scale grows
exponentially with increasing orbit separation, with billionyear stability for our Solar System requiring k * 13. Furthermore, simulations of the stability of test particles in
the current Solar System (Holman 1997) seem to show
that there remain few stable orbits in the outer Solar System other than those near Trojan points. This provides
circumstantial evidence that a small value of k is not
enough to separate stable orbits, since the outer planets
are separated from each other by more than 15 Hill radii.
For these reasons, our experiments use several radius-exclusion laws, including various combinations of Hill radii,
2:3 resonances, eccentricities, and k.
It is easy to see why radius-exclusion laws tend to produce planetary distances that approximately follow a geometric progression. If a fixed fractional radius exclusion V
is used for every planet, and planets are packed as tightly
as possible according to the radius-exclusion law, then the
physical extent of radius exclusion at distance r is rV, and
the resulting planetary separations would follow an exact
geometric progression with semi-major axis ratio (1 1 V)/
(1 2 V). If the planets are packed less tightly, then noise
is added to the fit.
2.2. Generating and Fitting Planetary Systems
Assume that planet i has a fractional radius exclusion
of Vi . To construct a sample system that satisfies the radiusexclusion law, we generate a list of nine planet distances
distributed uniformly random in log r between 0.2 and 50
AU and then sort them into increasing order hri ,
ri11j7i50 . If the list does not satisfy
ri11 2 ri . Vi11 1 Vi,
i 5 0, ... , 7,
then the entire list is discarded and we start over. The
‘‘trials’’ column of Table I lists the average number of such
trials that are required to generate a sample hrij8i50 satisfying
the described radius-exclusion law. We then generate 4096
such samples for each of the radius-exclusion laws described in Table I. For ei , we use the maximum eccentricity
for each planet in our Solar System over the past 3 million
years (Quinn et al. 1991). Not surprisingly, the mean number of trials needed to build a ‘‘valid’’ system that satisfies
the radius-exclusion criterion increases as the exclusion
radii get larger. Our most stringent exclusion criterion is
8Hi , which demands that adjacent planets are separated
by 8 times the sum of their respective Hill radii. We did
not test more stringent cases because they would be prohibitively expensive in terms of computer time (cf. the ‘‘trials’’ column).
For each sample that satisfies the relevant radius-exclusion, we perform a nonlinear least-squares fit of the distances ri to
a 1 bc i,
(3)
which we call a ‘‘generalized Bode law.’’ The fit is performed by minimizing the objective function
x2 5
O Slog(a 1 bcs ) 2 log r D ,
8
i
i 50
i
2
i
(4)
constrained so that a, b . 0 and c . 1. We fit on log r
rather than r because we want the fractional error of each
planet to be weighted equally. In the Appendix, we analytically derive approximations to the standard deviations si.
The initial guess for the parameters in the objective function is
c0 5
O
1 7 ri11
,
8 i50 ri
b0 5 r8 /c80 ,
a0 5 max(0, r0 2 b0).
To more accurately reflect the various forms of Bode’s
law, we also attempt fits that ‘‘ignore’’ 1, 2, and 3 planets.
We do this by performing fits on all (9j ), j 5 0, 1, 2, 3,
possible combinations of ignoring j out of 9 planets, and
choosing the best fit for each j. We then repeat the entire
procedure, allowing one gap to be inserted between the
two adjacent planets with the largest ri11/ri ratio, to mimic
the gap between Mars and Jupiter. This gives us 8 fits out
of 2 o3j50 ( 9j ), 5 260 combinations of ways to ignore planets
for each sample planetary system.
3. RESULTS
3.1. x 2 Fits
Results of all the fits for each type of system are presented in Figs. 1 and 2. Not surprisingly, the best fit for
552
HAYES AND TREMAINE
FIG. 1. x 2 fits, using Eq. (4). A system with a ‘‘gap’’ has had a virtual planet inserted between the two real planets that are most widely spaced
in log r, to mimic the gap between Mars and Jupiter in our Solar System. The x 2 of our own Solar System is labeled with ‘‘1’’ symbols, and the
means of 4096 random ones are labeled with diamonds. The labels on the horizontal axis correspond to the names described in Table I.
our own Solar System always occurs when a gap is added
between Mars and Jupiter, while Mercury, Neptune, and
Pluto are ignored. Even for our own Solar System, the fit
depends slightly on the radius-exclusion law, since this
affects the denominator si (see Appendix); for the three
cases in which the radius exclusion for each planet is identical in log r (M0, MJ, 2:3), the best fit for our Solar System
is identically
0.450 1 0.132 3 2.032i,
i 5 0, 1, ... , 8,
with x 2 values of 0.003, 0.005, and 0.009, respectively. This
result can be compared to the original Bode’s law, Eq. (1).
Consider the case where no gaps are allowed, and up
to three planets may be ignored (top row of Fig. 1). When
there is no radius-exclusion law (left edge of each figure),
the x 2 for the Solar System is always substantially less (by
a factor 3–6) than the mean of the random systems with
the same number of ignored planets; this is consistent with
the conclusion that the Solar System satisfies a generalized
Bode law significantly better than ones that are uniformly
random in log r. However, as we apply more stringent
radius-exclusion laws (moving right in each figure), the x 2
values for the Solar System become quite similar to the
mean of the random systems, indicating that the Solar
System is no closer to a generalized Bode law than random
ones that satisfy radius exclusion. This situation changes
in the bottom row of Fig. 1, which shows the case where
a gap is allowed. In particular, in the case with three planets
ignored and one gap, our Solar System’s best x 2 value is
TABLE I
The Various Radius-Exclusion Laws Used
Name
Description
Trials
M0
1Hi
2Hi
MJ
4Hi
Hei
2:3
23e
8Hi
Each planet has zero mass
Hill radii corresponding to planets of our Solar System
Like 1Hi , except radius exclusion of 2 Hill radii
All planets have Jupiter’s fractional Hill radius
Like 1Hi , except radius exclusion of 4 Hill radii
Adjacent planets no closer than Hi 1 ei
Adjacent planets no closer than the 2:3 resonance
Adjacent planets no closer than max(Hi 1 ei , H2:3)
Like 1Hi , except radius exclusion of 8 Hill radii
1
1.92
3.94
7.88
19.3
44.1
230
479
2820
Ex R LID(%)
0.4
0.8
1.6
1.9
4.5
3.4
10.5
9.7
11.7
LID R Ex(%)
100
93.3
87.2
49.3
57.6
23.1
24.8
16.9
1.7
Note. The table is ordered by the ‘‘trials’’ column, which is the 4096-sample average number of Monte-Carlo trials
required from a log-uniform distribution to find a sample that satisfies the corresponding radius exclusion criterion. The
last two columns (see Section 4) compare agreement between exclusion laws and the law of increasing differences (LID)
for a nine-planet system: specifically, how often the listed exclusion law produces a system satisfying LID (second-last
column), and vice versa (last column).
RANDOM PLANETARY SYSTEMS AND TITIUS–BODE LAWS
553
FIG. 2. The quantile of the Solar System’s x 2, i.e., the fraction of the 4096 random systems that have a x 2 better (i.e., smaller) than that of the
Solar System. We include the cases both with and without a gap (‘‘G’’ and ‘‘N’’, respectively). The labels on the horizontal axis correspond to the
names described in Table I.
consistently 1–1.5 orders of magnitude smaller than the
mean of the random systems. We suggest that this is because the particular exceptions we investigated were historically designed specifically to make our Solar System fit
better. In a different planetary system, other exceptions
or even entirely different (and arbitrary) exception rules
could be envisaged; for example, the planets could be split
into two groups, one group satisfying one rule, while the
other group satisfies another. The possibilities for inventing
exception rules are essentially endless.
Figure 2 shows the Solar System’s quantile—the fraction
of random systems with the same exceptions that have a
x 2 better (i.e., smaller) than that of the Solar System. A
small quantile would indicate that the Solar System fits
Bode’s law better than most random systems. Our Solar
System’s quantile is not exceptional if no gap is allowed,
ranging from about 0.15 to 0.7, and only mildly exceptional
if a gap is allowed (*0.04), and generally becomes less
exceptional as the random planetary systems are chosen
with increasingly stringent radius-exclusion laws (moving
right in each figure). The rightmost case, labeled 8Hi , gives
anomalously low quantiles in several cases. This probably
reflects the fact that our evaluation of the normalization
of x 2 (see Appendix) is only approximate when the planets
have different radius exclusions, and the approximation
worsens with increasing exclusion.
One could also argue that the main asteroid belt should
‘‘count’’ as an object. Figure 3 shows the Solar System’s
quantile for this case, with no exceptions allowed. In this
case, the Solar System’s x 2 value shows that it fits Bode’s
law better than 90%–98% of the random systems. However,
this result is not particularly significant, given that the case
we are examining is still, to some extent, tailored to the
properties of our Solar System.
increasing the strength of the radius exclusion shifts the
distribution to the right, because we are forcing more space
between planets. The peak of the histogram is sharpest
with the strongest radius exclusion (8Hi) and no exceptions,
because the regions allowed to contain planets are tightly
squeezed by the exclusion. If we allow a gap (cf. the two
bottom figures compared to the two top ones), Eq. (3)
effectively models an extra planet, so the distribution shifts
slightly toward smaller c values. Finally, increasing the
number of exceptions (cf. the two right figures compared
to the two left ones) causes the distribution to spread out
and flatten (because there are fewer constraints), and shift
to the right (because the planets that get ignored most
often are the inner and outer ones—cf. Fig. 6). One should
not read too much into the fact that the values of c for
stringent radius exclusion laws tend to cluster around the
value of 2.0, which is the value for our own Solar System.
This value is biased by our decision to fit 9 planets between
0.2 and 50 AU, because (50/0.2)1/8 5 1.99.
Scatter plots of b vs a are presented in Fig. 5. The most
obvious feature is that most systems appear above the line
a 1 b P 0.2; this is expected because 0.2 is the smallest
orbital radius that we allow. As we increase the exclusion
from Hi to 8Hi (cf. the two bottom figures compared to
the two top ones), the scatter decreases for the same reason
3.2. Distributions of a, b, and c
Although our chief concern in this paper is with how
well random planetary systems fit Eq. (3), we can also
discuss the values of the fitting parameters a, b, and c that
we obtained.
The distribution of c is shown in Fig. 4. In general,
FIG. 3. The quantile of the Solar System, compared to 4096 random
ones, when no exceptions or gaps are allowed, but the main asteroid belt
is included as a ‘‘planet’’ at radius 2.8 AU.
554
HAYES AND TREMAINE
FIG. 4. Normalized histogram of the value of the fitting parameter c from Eq. (3). In each of these figures, the number of exceptions (i.e., the
number of planets that are ignored in the fit) and the number of gaps (always 0 or 1) are held constant while we plot the distribution for various
radius-exclusion laws as discussed in Section 2.1. The graphs that have 1 and 2 exceptions can be well approximated by interpolating between the
left and the right figures.
the distribution of c becomes more peaked in Fig. 4: the
systems are more tightly constrained by the exclusion law.
The value of the parameter a in Eq. (3) is zero in about
25% of the samples, in which case Bode’s law corresponds
to a geometric sequence.
Finally, there was no observable correlation of a, b, or
c with x 2.
3.3. Other Observations
A histogram of which planet gets ignored when there is
one exception and no radius exclusion is shown in Fig. 6.
The innermost and outermost planets are ignored most
often, which is expected since a planet with only one neighbor is less constrained than those with two.
The placement of the gap is approximately uniform between all planet pairs, for all types of systems. Adding a
gap produced a better fit in about 65% of all cases.
It is prudent to show that our results are not strongly
dependent upon the assumption that the underlying distribution is uniform in log r. If instead we assume a disk
surface density that scales as r23/2, which roughly corresponds to the expected density in the protoplanetary disk
(Lissauer 1993), then if all the planets are equally massive,
they should be uniformly distributed in Ïr. We therefore
performed our entire suite of experiments again, this time
trying to fit an a 1 bc i law to planetary systems with an
underlying random distribution that is uniform in Ïr. We
find that most of the above results are qualitatively unchanged. For example, in the case with no exceptions and
no gap, the fit of the random systems worsens, so that our
Solar System’s quantile gets better, but only by about 0.05
to 0.15. In the case of three exceptions and a gap, the Solar
System’s quantile is almost the same in the Ïr distribution
as in the log r distribution. Furthermore, as radius exclusion
increases, the effect of the underlying distribution is suppressed because radius exclusion is biased toward accepting planetary systems that follow a roughly geometric
progression. We conclude that our comparisons are not
strongly affected by the assumption that the underlying
distribution is uniform in log r.
4. THE LAW OF INCREASING DIFFERENCES
Efron (1971) and Conway and Zelenka (1988) have
noted that the distance between planets in the Solar System
RANDOM PLANETARY SYSTEMS AND TITIUS–BODE LAWS
555
FIG. 5. Scatter plots of the fitting parameters b and a from Eq. (3). In Figs. i–iv, a is exactly zero in 47, 26, 47, and 5% of the cases, respectively.
is an increasing function of distance for all adjacent pairs
except Neptune–Pluto; the ‘‘major’’ satellites of Jupiter,
Saturn and Uranus also satisfy this relation (Conway and
Zelenka 1988). They propose that this ‘‘law of increasing
differences’’ is a reasonable null hypothesis to use when
testing the statistical significance of Bode’s law. They note
that a pure log-uniform distribution produces increasing
differences only a small percentage of the time and that if
FIG. 6. A histogram showing the fraction of systems in which planet
i is ignored, as a function of i, when one exception is allowed and there
is no radius exclusion. The distribution does not change significantly in
other systems.
we assume the law of increasing differences then the success of Bode’s law is unsurprising.
We are uncomfortable with this law because it has no
physical basis. For example, there is no dependence on
planetary mass, which is unrealistic. We also note that the
law does not apply to all satellites around any planet and
that there is no natural way to define what constitutes a
‘‘major’’ satellite. This concern has prompted us to examine the relation between radius-exclusion laws and the law
of increasing differences. Table I shows the occurrences
of agreement between the law of increasing differences
and radius exclusions. As Table I shows, (i) a system that
satisfies radius exclusion rarely satisfies the law of increasing differences; (ii) one that satisfies the law of increasing
differences will often satisfy all but the most stringent exclusion laws. We also observed that (iii) the number of
trials required to find a random log-uniform sample that
satisfies the law of increasing differences is 1 to 2 orders
of magnitude larger than that for radius exclusion; (iv)
random planetary systems that satisfy the law of increasing
differences have x 2 values that are 1.3 to 3 times smaller
than ones generated using radius exclusion.
For these reasons, we believe that the law of increasing
differences is a much more restrictive assumption than
radius-exclusion laws, and in the absence of any physical
556
HAYES AND TREMAINE
justification, it does not form a sound basis from which to
judge the validity of Bode’s law.
5. DISCUSSION AND CONCLUSIONS
We have measured the deviation from Bode’s law of
planetary systems whose distances are distributed uniformly random in log r, subject to radius exclusion constraints. We find that, as radius exclusion becomes more
stringent, the systems tend to fit Bode’s law better. We
compare these fits to that of our own Solar System. We
find that, when no exceptions or gaps are allowed, our
Solar System fits marginally better than random systems
that follow weak radius-exclusion laws, but fits no better,
or even worse, than those that satisfy more stringent but
still reasonable radius exclusions. If one gap is allowed to
be added, and up to three planets are ignored, then our
Solar System fits significantly better than random ones
built with weak radius exclusions and marginally better
than ones with strong radius exclusions; however, this modest success for Bode’s law probably arises because these
rules (three planets removed, one gap added) were designed specifically in earlier centuries to make our Solar
System fit better.
Even though our underlying distance distribution, uniform in log r, is scale invariant, the analysis of Graner and
Dubrulle (1994) and Dubrulle and Graner (1994) does
not apply to most of our cases, since the distribution of
planetary masses and Hill radii is not scale-invariant. Further, we found that, even in the cases where the radiusexclusion law is scale-invariant (M0 , MJ , 2:3), the bestfitting generalized Bode law has a ? 0 in 36% of the cases
and hence is not scale-invariant.
Our approach to Bode’s law is very simplistic. We ignore
all of the details of planet formation (secular resonances,
planet migration, resonance capture, chaos, etc.) and use
simple stability criteria to discard unstable systems. The
natural next step is to replace the approximate stability
criteria we have used by actual orbit integrations. We conjecture that if we repeat the experiments of the present
paper using a direct integration of several Gyr (e.g., Wisdom and Holman 1991) to discard unstable systems, we
would find that the surviving systems fit a generalized Bode
law better than the random ones in this paper and approximately as well as our own Solar System. This would
strengthen our conclusion that the significance of Bode’s
law is simply that stable planetary systems tend to be regularly spaced.
APPENDIX: MEAN, VARIANCE, COMPRESSION OF
UNIFORM DISTRIBUTIONS
If n numbers are chosen from the uniform random interval U[0, 1] and
sorted into nondecreasing order, then the mean and variance of the ith
one Xi are (Rice 1988, problem 4.15)
Xi 5 i/(n 1 1),
s 2Xi 5
i(n 2 i 1 1)
,
(n 1 1)2(n 1 2)
i 5 1, .... , n,
(5)
which we then scale to the interval [log(0.2), log(50)] used in this paper.
Radius exclusion makes the allowed intervals between planets smaller,
thus compressing the uniform standard deviations, Eq. (5), by a factor
Ci . To compute Ci , assume ri 5 1 and that all the planets have the same
fractional Hill radius H and perfectly fit a Bode’s law with exponent
c 5 b and a 5 0. It is easy to show that
Ci 5
log(b)
,
log(b) 1 log(1 2 H) 2 log(1 1 H)
(6)
finally giving
s 2i 5 (s xi /Ci )2,
(7)
which we substitute into Eq. (4). When distance is measured in log r and
all planets have the same radius exclusion, Eq. (6) is exact; otherwise it
is only approximate.
ACKNOWLEDGMENTS
S.T. thanks Gerald Quinlan for many discussions on this and related
subjects. W.H. thanks his supervisor, Professor Ken Jackson, for moral
support for the duration of this research, which was conducted during,
but not related to, his doctoral work. We thank Rosemary McNaughton,
François Pitt, John E. Chambers and an anonymous referee for constructive comments. S.T. acknowledges support from an Imasco Fellowship.
This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada and the Information Technology Research
Centre of Ontario.
REFERENCES
Birn, J. 1973. On the stability of the planetary system. Astron. Astrophys.
24, 283–293.
Chambers, J. E., G. W. Wetherill, and A. P. Boss 1996. The stability of
multi-planet systems. Icarus 119, 261–268.
Conway, B. A., and T. J. Elsner 1988. Dynamical evolution of planetary
systems and the significance of Bode’s Law. In Long-Term Dynamical
Behavior of Natural and Artificial N-body Systems (A. E. Roy, Ed.),
pp. 3–12. Kluwer Academic, Amsterdam.
Conway, B. A., and R. E. Zelenka 1988. Further numerical investigations
into the significance of Bode’s Law. In Long-Term Dynamical Behavior
of Natural and Artificial N-body Systems (A. E. Roy, Ed.), pp. 13–20.
Kluwer Academic, Amsterdam.
Dubrulle, B., and F. Graner 1994. Titius–Bode laws in the Solar System. II.
Build your own law from disk models. Astron. Astrophys. 282, 269–276.
Efron, B. 1971. Does an observed sequence of numbers follow a simple
rule? (Another look at Bode’s Law). J. Am. Statist. Assoc. 66(335),
552–559.
Fernández, J. A., and W.-H. Ip 1984. Some dynamical aspects of the
accretion of Uranus and Neptune—The exchange of orbital angular
momentum with planetesimals. Icarus 58, 109–120.
Gladman, B. 1993. Dynamics of systems of two close planets. Icarus
106, 247–263.
Good, I. J. 1969. A subjective evaluation of Bode’s Law and an ‘‘objective’’ test for approximate numerical rationality. J. Am. Statist. Assoc.
64, 23–66.
RANDOM PLANETARY SYSTEMS AND TITIUS–BODE LAWS
Graner, F., and B. Dubrulle 1994. Titius–Bode laws in the Solar System. I.
Scale invariance explains everything. Astron. Astrophys. 282, 262–268.
Hénon, M. 1969. A comment on ‘‘The Resonant Structure of the Solar
System’’ by A. M. Molchanov. Icarus 11, 93–94.
Hills, J. G. 1970. Dynamic relaxation of planetary systems and Bode’s
Law. Nature 225, 840–842.
Holman, M. J. 1997. The distribution of mass in the Kuiper Belt. Preprint,
Canadian Institute of Theoretical Astrophysics.
Holman, M. J., and N. W. Murray 1996. Chaos in high-order mean motion
resonances in the outer asteroid belt. Astron J. 112(3), 1278–1293.
Ipatov, S. I. 1993. Migration of bodies in the accretion of planets. Solar
Sys. Res. 27, 65–79.
Laskar, J. 1997. Large scale chaos and the spacing of the inner planets.
Astron. Astrophys. 317, L75–L78.
Lecar, M. 1973. Bode’s Law. Nature 242, 318–319.
Li, X. Q., H. Zhang, and Q. B. Li 1995. Self-similar collapse in nebular
disk and the Titius–Bode law. Astron. Astrophys. 304, 617–621.
Lin, D. N. C., P. Bodenheimer, and D.C. Richardson 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location.
Nature 380, 606–607.
Lissauer, J. 1987. Timescales for planetary accretion and the structure
of the protoplanetary disk. Icarus 69, 249–265.
Lissauer, J. 1993. Planet formation. Annu. Rev. Astron. Astrophys. 31,
129–174.
Llibre, J., and C. Piñol 1987. A gravitational approach to the Titius–Bode
Law. Astron J. 93(5), 1272–1279.
Malhotra, R. 1993. The origin of Pluto’s peculiar orbit. Nature
365(6499), 819–821.
Malhotra, R. 1995. The origin of Pluto’s orbit: Implications for the Solar
System beyond Neptune. Astron J. 110, 420.
557
Molchanov, A. M. 1968. The resonant structure of the Solar System: The
Law of Planetary Distances. Icarus 8, 203–215.
Nieto, M. M. 1972. The Titius–Bode Law of Planetary Distances: Its
History and Theory. Pergamon, Oxford, New York.
Patterson, C. W. 1987. Resonance capture and the evolution of the planets.
Icarus 70, 319–333.
Quinn, T. R., S. Tremaine, and M. Duncan 1991. A three million year
integration of the Earth’s orbit. Astron J. 101(6), 2287–2305.
Rice, J. A. 1988. Mathematical Statistics and Data Analysis. Wadsworth &
Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
Roy, A. E. (Ed.) 1988. Long-Term Dynamical Behaviour of Natural and
Artificial N-Body Systems. Kluwer Academic, Amsterdam.
Trilling, D. E., W. Benz, T. Guillot, J. I. Lunin, W. B. Hubbart, and
A. Burrows 1998. Orbital evolution and migration of giant planets:
Modelling extrasolar planets. Astrophys. J., in press.
Weidenschilling, S. J., and D. R. Davis 1985. Orbital resonance in the
solar nebula: Implications for planetary accretion. Icarus 62, 16–29.
Wetherill, G. W. 1988. Accumulation of Mercury from Planetesimals. In
Mercury (F. Vilas, C. R. Chapman, and M. S. Matthews, Eds.), pp.
670–691. Univ. of Arizona Press, Tucson.
Wetherill, G. W., and L. P. Cox 1984. The range of validity of the twobody approximation in models of terrestrial planet accumulation. I.
Gravitational perturbations. Icarus 60, 40–55.
Wetherill, G. W., and L. P. Cox 1985. The range of validity of the twobody approximation in models of terrestrial planet accumulation. II.
Gravitational cross sections and runaway accretion. Icarus 63, 290–303.
Wisdom, J. 1980. The resonance overlap criterion and the onset of stochastic behavior in the restricted three-body problem. Astron J. 85(8), 1122–
1133.
Wisdom, J., and M. Holman 1991. Symplectic maps for the n-body problem. Astron J. 102, 1528–1538.